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We propose a novel estimation approach for a general class of semi-parametrictime series models where the conditional expectation is modeled through aparametric function. The proposed class of estimators is based on a Gaussianquasi-likelihood function and it relies on the specification of a parametricpseudo-variance that can contain parametric restrictions with respect to theconditional expectation. The specification of the pseudo-variance and theparametric restrictions follow naturally in observation-driven models withbounds in the support of the observable process, such as count processes anddouble-bounded time series. We derive the asymptotic properties of theestimators and a validity test for the parameter restrictions. We show that theresults remain valid irrespective of the correct specification of thepseudo-variance. The key advantage of the restricted estimators is that theycan achieve higher efficiency compared to alternative quasi-likelihood methodsthat are available in the literature. Furthermore, the testing approach can beused to build specification tests for parametric time series models. Weillustrate the practical use of the methodology in a simulation study and twoempirical applications featuring integer-valued autoregressive processes, whereassumptions on the dispersion of the thinning operator are formally tested, andautoregressions for double-bounded data with application to a realizedcorrelation time series.

Pseudo-variance quasi-maximum likelihood estimation of semi-parametric time series models